The Accumulation Function - Classwork

Transcription

1 The Accumulation Function - Classwork Now that we understand that the concept of a definite integral is nothing more than an area, let us consider a very special type of area problem. Suppose we are given a function f # 2. We know that to be a horizontal line at # 2 is the same as f ( t) # 2 is the same as f ( k) # 2. y = 2. First, realize that the equation of the graph of f Whether we use, t, or k, it does not matter. The graph is still a horizontal line. We are used to using but we will see a good reason that we will occasionally be using another letter. f ( t) # 2 Consider the epression Above, we have the graph f t function of. As changes, # 2. We now want to consider the epression f ( t) dt changes as well. Complete the chart below.. What is this? It is a It should be apparent that as gets bigger, accumulating area and we call f ( t) # 2 to describe our line rather than f Let' s calculate the value of Complete the chart below increases as well. What we appear to be doing is the accumulation function. Finally, it should be obvious why we use # 2. f d f ( t) dt for # to 4 f t would be confusing MasterMathMentor.com Stu Schwartz

2 Eample 1) Let F # where the graph of f is below. Remember f is the same thing as f ( t). as the rate of snowfall over a period of time. For instance, at = 1, snow is falling Think of f at 3 inches per hour, at = 3, it is not snowing, and at = 4, snow is melting at 4 inches per hour. y # f a. Complete the chart. In the snow analogy, F represents the accumulation of snow over time F d d b. Now let s consider F" # happen? So F" # Knowing that, let s complete the chart.. If we take the derivative of an integral, what would you epect to d d is the same thing as F" c. On what subintervals of [, 8] is F increasing? Decreasing? d. Where in the interval [, 8] does F achieve its minimum and maimum value? What are those values? f. Find the concavity of F and any inflection points. Justify your answer h. Sketch a rough graph of F MasterMathMentor.com Stu Schwartz

3 Eample 2) Let F # where f is the function graphed below (consisting of lines and a semi-circle) y # f Find the following: a) F b) F( 2) c) F( 4) d) F( 6) e) F (# ) 1 f) F # 2 g) F # 3 h) F # 4 i) F"( 4) j) F"( 2) k) " F 6 l) F" # 3 m) On what subintervals of # 4, 6 is F increasing and decreasing. Justify your answer. n) Where in the interval # 4, 6 does F achieve its minimum value? What is the minimum value? o) Where in the interval # 4, 6 does F achieve its maimum value? What is the minimum value? p) Where on the interval # 4, 6 is F concave up? Concave down? Justify your answer. q) Where does F have points of inflection? r) Sketch the function F. each tic mark on y-ais is 2 units MasterMathMentor.com Stu Schwartz

4 The Accumulation Function - Homework y # f 1. Let F # where the graph of f is above (the graph consists of lines and a quarter circle) a. Complete the chart F " F b. On what subintervals of [, 8] is F increasing? Decreasing? Justify your answer. c. Where in the interval [, 8] does F achieve its minimum value? What is the minimum value? Justify answer. d. Where in the interval [, 8] does F achieve its maimum value? What is the maimum value? Justify answer. e. Find the concavity of F and any inflection points. Justify answers. f. Sketch a rough graph of F MasterMathMentor.com Stu Schwartz

5 y # f 2. Let F # where the graph of f is above (the graph consists of lines and a semi-circle) a. Complete the chart F " F b. On what subintervals of # 4, 8 is F increasing? Decreasing? c. Where in the interval # 4, 8 does F achieve its minimum value? What is the minimum value? Justify answer. d. Where in the interval # 4, 8 does F achieve its maimum value? What is the maimum value? Justify answer. e. On what subintervals of # 4, 8 is F concave up and concave down? Find its inflection points. Justify answers. f. Sketch a rough graph of F MasterMathMentor.com Stu Schwartz

6 The Accumulation Function - Life Application You have bought 1 shares of XYZ stock and decide to keep it for 15 days. Below is a graph that represents the change of the price of the your stock on each day. For instance, on the end of day 1, the stock has increased by $1 a share. At the end of day 4, it has not changed. At the end of day 6, it has gone down by $3.5 a share. We will call the graph you see below f ( t). made up of straight lines. It is obviously a function of time. Remember that the function represents the change in the value of your stock, not the value of the stock Let F #. Answer the following questions: 1. Complete the chart below f F 2. What is the real life meaning of F? 3. For what intervals is F increasing and decreasing. Justify your answer. 4. In the interval [, 15], find the minimum value of F and the day it is reached. Justify your answer. 5. In the interval [, 15], find the maimum value of F and the day it is reached. Justify your answer. 6. Determine the concavity of F and justify your answer. MasterMathMentor.com Stu Schwartz

7 7. Below, sketch F. 8) Based on your findings, find: a) how much money you made(lost) on XYZ stock in that 15 day time period. b) the day that the stock s value had the biggest rise c) the days between which the stock s value had the steepest rise (not the same question) d) the day that the stock s value had the biggest decline e) the days between which the stock s value had the steepest decline (not the same question) f) the day you wished you sold your stock g) the day you are glad you didn t sell your stock MasterMathMentor.com Stu Schwartz